Normal form theory for relative equilibria and relative periodic solutions
HTML articles powered by AMS MathViewer
- by Jeroen S. W. Lamb and Ian Melbourne PDF
- Trans. Amer. Math. Soc. 359 (2007), 4537-4556 Request permission
Abstract:
We show that in the neighbourhood of relative equilibria and relative periodic solutions, coordinates can be chosen so that the equations of motion, in normal form, admit certain additional equivariance conditions up to arbitrarily high order. In particular, normal forms for relative periodic solutions effectively reduce to normal forms for relative equilibria, enabling the calculation of the drift of solutions bifurcating from relative periodic solutions.References
- V. I. Arnol′d, Geometrical methods in the theory of ordinary differential equations, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 250, Springer-Verlag, New York, 1988. Translated from the Russian by Joseph Szücs [József M. Szűcs]. MR 947141, DOI 10.1007/978-1-4612-1037-5
- Peter Ashwin and Ian Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity 10 (1997), no. 3, 595–616. MR 1448578, DOI 10.1088/0951-7715/10/3/002
- David Chan, Hopf bifurcations from relative equilibria in spherical geometry, J. Differential Equations 226 (2006), no. 1, 118–134. MR 2232432, DOI 10.1016/j.jde.2005.09.015
- D. Chan and I. Melbourne. A geometric characterisation of resonance in Hopf bifurcation from relative equilibria. Preprint, 2006.
- A. Comanici. Transition from rotating waves to modulated rotating waves on the sphere. SIAM J. Applied Dynamical Systems 5 (2006) 759–782.
- C. Elphick, E. Tirapegui, M. E. Brachet, P. Coullet, and G. Iooss, A simple global characterization for normal forms of singular vector fields, Phys. D 29 (1987), no. 1-2, 95–127. MR 923885, DOI 10.1016/0167-2789(87)90049-2
- Bernold Fiedler, Björn Sandstede, Arnd Scheel, and Claudia Wulff, Bifurcation from relative equilibria of noncompact group actions: skew products, meanders, and drifts, Doc. Math. 1 (1996), No. 20, 479–505. MR 1425301
- Bernold Fiedler and Dmitry Turaev, Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions, Arch. Ration. Mech. Anal. 145 (1998), no. 2, 129–159. MR 1664546, DOI 10.1007/s002050050126
- M. J. Field. Symmetry Breaking for Compact Lie Groups. Memoirs of the Amer. Math. Soc. 574, Amer. Math. Soc., Providence, RI, 1996.
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- Martin Golubitsky, Ian Stewart, and David G. Schaeffer, Singularities and groups in bifurcation theory. Vol. II, Applied Mathematical Sciences, vol. 69, Springer-Verlag, New York, 1988. MR 950168, DOI 10.1007/978-1-4612-4574-2
- John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1990. Revised and corrected reprint of the 1983 original. MR 1139515
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR 499562
- Jeroen S. W. Lamb, Local bifurcations in $k$-symmetric dynamical systems, Nonlinearity 9 (1996), no. 2, 537–557. MR 1384491, DOI 10.1088/0951-7715/9/2/015
- Jeroen S. W. Lamb and Ian Melbourne, Bifurcation from discrete rotating waves, Arch. Ration. Mech. Anal. 149 (1999), no. 3, 229–270. MR 1726677, DOI 10.1007/s002050050174
- J. S. W. Lamb, I. Melbourne, and C. Wulff, Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions, J. Differential Equations 191 (2003), no. 2, 377–407. MR 1978383, DOI 10.1016/S0022-0396(03)00019-6
- Jeroen S. W. Lamb and John A. G. Roberts, Time-reversal symmetry in dynamical systems: a survey, Phys. D 112 (1998), no. 1-2, 1–39. Time-reversal symmetry in dynamical systems (Coventry, 1996). MR 1605826, DOI 10.1016/S0167-2789(97)00199-1
- Jeroen S. W. Lamb and Claudia Wulff, Reversible relative periodic orbits, J. Differential Equations 178 (2002), no. 1, 60–100. MR 1878526, DOI 10.1006/jdeq.2001.4004
- Mark Roberts, Claudia Wulff, and Jeroen S. W. Lamb, Hamiltonian systems near relative equilibria, J. Differential Equations 179 (2002), no. 2, 562–604. MR 1885680, DOI 10.1006/jdeq.2001.4045
- Floris Takens, Forced oscillations and bifurcations, Applications of global analysis, I (Sympos., Utrecht State Univ., Utrecht, 1973) Comm. Math. Inst. Rijksuniv. Utrecht, No. 3-1974, Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974, pp. 1–59. MR 0478235
- Claudia Wulff, Transitions from relative equilibria to relative periodic orbits, Doc. Math. 5 (2000), 227–274. MR 1758877
- Claudia Wulff, Jeroen S. W. Lamb, and Ian Melbourne, Bifurcation from relative periodic solutions, Ergodic Theory Dynam. Systems 21 (2001), no. 2, 605–635. MR 1827120, DOI 10.1017/S0143385701001298
- Claudia Wulff and Mark Roberts, Hamiltonian systems near relative periodic orbits, SIAM J. Appl. Dyn. Syst. 1 (2002), no. 1, 1–43. MR 1893733, DOI 10.1137/S1111111101387760
Additional Information
- Jeroen S. W. Lamb
- Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
- MR Author ID: 319947
- Email: jeroen.lamb@imperial.ac.uk
- Ian Melbourne
- Affiliation: Department of Mathematics and Statistics, University of Surrey, Guildford GU2 7XH, United Kingdom
- MR Author ID: 123300
- Email: ism@math.uh.edu
- Received by editor(s): November 15, 2005
- Published electronically: April 17, 2007
- Additional Notes: The first author would like to thank the UK Engineering and Physical Sciences Research Council (EPSRC), the Nuffield Foundation and the UK Royal Society for support during the course of this research.
The first and second authors would like to thank IMPA (Rio de Janeiro) for hospitality during a visit in which part of this work was done. - © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 4537-4556
- MSC (2000): Primary 37G40, 37G05, 37G15, 37C55
- DOI: https://doi.org/10.1090/S0002-9947-07-04314-0
- MathSciNet review: 2309197